• Lab could go on very long
  • Some students connected the capacitor in parallel due to the design of the connector

Circuits II: Capacitance

In Part I this project, we explored resistors and DC circuits. For Part II, we will be looking a new type of circuit element – the capacitor – which is an important component in AC circuits. In order to use this element, we will also introduce the function generator – capable of producing time-dependent voltages like sine waves and square waves – and the oscilloscope – a tool useful for observing and measuring time-dependent voltages (kind of like an AC analog to the DC voltmeter used in Part I).


  • To use a function generator to create sine and square wave voltages;
  • To introduce the basic operations of an oscilloscope;
  • To introduce properties of capacitors;
  • To measure the RC time constant of different circuits; and
  • To observe the non-idialallity of measurement devices and power sources.

Lab notebook template

One member of the group should click on the link below to start your group lab notebook. (You may be asked to log into your UChicago Google account if you are not already logged in.) Make sure to share the document with everyone in the group (click the “Share” button in the top right corner of the screen) so each member has access to the notebook after you leave lab.

Digital oscilloscope

The digital oscilloscope (hereafter referred to simply as a digital scope) is a measurement tool which is very commonly used in the physics laboratory. Here we provide some details on the basic functions and operation of digital scopes. You will then make use of them to observe and measure time varying voltages in some capacitor circuits.

Principle of operation

The oscilloscope is a fast voltmeter which can be used to give a graphical display of voltage versus time or voltage versus voltage.

Voltages are applied to the central pins of BNC connectors on the front panel of the scope. The shell of the BNC is held at ground potential via the ground pin of the scope’s AC line cord. The signal voltage is measured (sampled) many times each second. Each measurement of voltage and time is displayed as a lit pixel, sweeping from left to right across the display. The line produced on the scope is often called the scope’s trace.

In order to produce a stable trace, the sweep must be in sync with the test signal. This synchronization is achieved by triggering the scope. In order to obtain proper triggering, the user must make some adjustments on the scope. Some useful terms are defined in the next section.


  • Triggering determines when the oscilloscope draws a signal on the screen. The trace will appear stationary if the scope is triggered at the same point in each cycle of the signal. That point on the signal at which triggering occurs may be specified by picking a particular value of voltage, slope, and offset as shown in Fig. 1.
  • The trigger level sets the magnitude of the voltage necessary to initiate a sweep.
  • The toggle between slope: rising or slope: falling sets the slope of the leading edge of the signal on which the scope is to trigger.
  • Adjusting the horizontal position will move the entire signal left or right on the screen.
Figure 1: A sine wave displayed on a scope that is set to trigger when the signal is $0 \text{ V}$ and falling.
What should changing these parameters look like?

As an illustration, Fig. 2. shows how changing the trigger settings can alter how the oscilloscope shows the exact same signal.  Note that the signal is always centered where the Horizontal offset and Trigger Level arrows intersect 

Figure 2: Examples of how changing triggering settings will change how the same signal is displayed.

There are many additional parameters that can be altered to change how a signal is displayed, such as:

  • Trigger source selects which input will receive the desired triggering signal.
    • Internal triggering tells the scope to use the signal applied to channel 1 or 2 as the trigger signal.
    • External triggering tells the scope that the trigger signal is being applied to the EXT input.
    • Line triggering tells the scope to use the $60\text{ Hz}$ power line as a trigger signal.
  • AC coupling places a capacitor in series with the input to remove any DC component from the signal. (DC coupling has no capacitor.)
  • The vertical scale sets the voltage scale of the vertical axis.
  • The horizontal scale sets the scale of the horizontal axis (which is usually time).
  • Menu buttons display options to be selected by the column of buttons at the right side of the display.

Scope exercises

Work through the following set of exercises with a focus on trying to understand how to use the digital scope and how the different controls affect the signal you are observing. The TA will check in with you periodically to verify your progress. You should be prepared to explain to the TA what is displayed on your scope screen and how the controls you have been using affect the signal.

Displaying a signal

Setup your function generator to output sine wave at a frequency of about $5\text{ kHz}$, make sure the amplitude is set near the max. Connect the function generator output to either input of the scope.

Now, figure out how to get the scope to display a stable trace showing a few cycles of the sine wave from the function generator. If you are having difficulty finding the signal, or obtaining a stable trigger feel free to consult with the TA or with other students in different groups.

Most commonly used oscilloscope controls:

  • The Softkeys are context dependent, and select whatever is adjacent to them on the screen.
  • The Multipurpose Knob is also context dependent; whenever the green LED next to it is lit up it will control a setting on the screen
    • The knob also acts as a button when pressed
  • The Trigger Menu brings up the triggering controls, which are then selected/changed with the softkeys
  • The Trigger Level knob sets the voltage that the oscilloscope triggers at
  • The Horizontal Scale knob changes the $x$ (time) axis of the scope.  This controls how many seconds (down to hundreds of nanoseconds) of data are shown on the screen at once.
  • The Vertical Scale knobs change the $y$ (voltage) axis for the two different inputs


Once you have obtained a stable trace on the scope go into the trigger menu and make sure that the trigger is set to Normal mode as opposed to Auto mode. Now vary the trigger level and try to understand what effect it has on the trace in regards to how the scope is triggering; how does the trigger level affect the stability of the trigger, when the scope is triggering stably how does raising or lowering the trigger affect the displayed waveform? Change the Slope setting in the trigger menu and determine how this changes the displayed waveform.

Try setting the trigger level near the top of the waveform. Now figure out how to vary the amplitude of the function generator output to make the scope stop triggering and then restore the triggering.

Make sure you understand how the amplitude of the waveform is interacting with the scope's trigger level to affect whether or not the scope is triggering. Can you reduce the amplitude of the waveform until the scope stops triggering and then make the scope trigger again using only the trigger level control on the scope?

If the amplitude of the signal under study varies significantly during measurements, triggering may be lost and re-adjustment of the trigger level can become tedious. In that case, another signal with large, constant amplitude, which is synchronous with the test signal may be used for more consistent External Triggering. The TTL output on the function generator provides such a signal.

Plug the TTL output from the function generator into the EXT input on the scope and set up the scope for external triggering.  Vary the function generator amplitude and ascertain how it affects the stability of the trace as compared to when you were triggering off of the sine wave itself.

What does TTL mean?

TTL stands for Transistor-Transistor Logic.  It is a set of standards for what range of voltages a computer (or other digital device) will recognize as being '0' ($0$-$0.8\text{ V}$) or '1' ($2$-$5\text{ V}$).  In practice, this output will generate a square wave that has the same frequency as our signal that alternates between $0$ and $5\text{ V}$.

Making measurements using the digital scope

The Cursor button will bring up various options which allow you to position cursors on the scope screen to make measurements of voltages and times.

Figure out how to use these features to measure the amplitude of your sine wave in volts and its period in seconds.

Make sure your TA has looked over your progress before continuing!



Defining capacitance

Consider two parallel conducting plates of area $A$ separated by a distance $d$. Such a device is called a capacitor. If we connect each plate to a voltage source like a battery or power supply, a charge will appear on each plate, as shown in Fig. 3.

Figure 3: Parallel plate capacitor connected to a voltage source

The amount of charge $q$ held in a capacitor depends on the capacitor’s capacitance $C$ and on the voltage $V$ applied as

$ q = CV$ $(1)$

It can be shown that the capacitance is given by

$C = \dfrac{\epsilon_0 A}{d}$, $(2)$

where $\epsilon_0 = 8.85418782\times 10^{-12}\,\,\dfrac{\mathrm{s}^4\mathrm{A}^2}{\mathrm{m}^3 \,\mathrm{kg}}$ is the permittivity of free space.

The SI unit of capacitance is the Farad. A capacitor is said to have a capacitance of one farad when one coulomb of charge produces a potential difference of one volt. The farad was considered an enormous unit for quite some time. It is more typical to find capacitors on the order of pico-, nano-, or microfarads. With improving material science, we now have developed what are termed supercapacitors that can be rated for 1,000's of Farads of capacitance.

Discharging a capacitor

If a resistor is connected across a charged capacitor by closing a switch as shown in Fig. 4, the excess electrons on the negative plate will move through the resistor to the positively charged (electron-deficient) plate. In other words, the capacitor will discharge.

Figure 4: Capacitor discharging circuit

Using Kirchhoff's rules, the voltage across the capacitor – which by $(1)$ is $V = q/C$ – plus the voltage across the resistor – which is $V=IR$ must sum to zero. Therefore,

$\dfrac{q}{C} -IR = 0$. $(3)$

Since $q$ and $I$ are related by the equation

$I = \dfrac{dq}{dt}$, $(4)$

we can substitute $(4)$ into $(3)$ and rearrange to find

$\dfrac{1}{q}dq = -\dfrac{1}{RC}dt$. $(5)$

Integrating this and putting in the initial condition $q(0) = q_0$, we find

$\displaystyle q(t) = q_0 e^{-t/RC}$, $(6)$

or, since by $(1)$, $q_0 = C V_0$ and $q=CV$,

$\displaystyle V_C(t) = V_0 e^{-t/RC}$. $(7)$

Thus, the voltage across the capacitor (or resistor) must decay exponentially. The time for the voltage to drop to $1/e \approx 37\%$ of its initial value will be when $t/RC = 1$, i.e., when $t=RC$. We define this time to be the time constant of a resistance-capacitance or “$RC$” circuit: $\tau = RC$

Eq. 7 assumes that the capacitor is discharging to 0V. If it this is not true (as will be the case with your function generator), we can generalize discharging behavior between an initial voltage $V_{max}$ and final voltage, $V_{min}$, to find the change in the capacitor voltage as

$\displaystyle \Delta V_C(t) = (V_{max}-V_{min}) e^{-t/RC}$. $(7b)$

where $\Delta V_C = V_C(t) - V_{min}$, i.e. the difference between the voltage across the capacitor and the minimum voltage it reaches. If $V_{min} = 0V$ this reduces right back to eq. 7.

As an example: if $V_{max} = 10 V$ and $V_{min} = -8V$, after one time constant $\tau$ has passed we would expect $\Delta V_C(t)$ voltage across the capacitor to be $(10V - -8V) * e^-1 \approx 6.6V$ So, we'd be looking for the time when the voltage we measure $V_C$ is at $V_{Min} + 6.6V = -1.4V$ to calculate the time constant.

A graph showing the prediction of $(7)$ for a discharging capacitor is shown in Fig. 5.

Figure 5: Discharge of a capacitor
Charging a capacitor

Suppose instead of starting with the capacitor charged, we start with it discharged and proceed to charge the capacitor through a resistor using a circuit such as that shown in Fig. 6.

Figure 6: Capacitor charging circuit

Using Kirchhoff's rules, $(1)$, and Ohm's law, the equation of the circuit is

$V_0 = IR + \dfrac{q}{C}$. $(8)$

Substituting $(4)$ for $I$, integrating, and putting in the boundary conditions, we find

$q(t) = CV_0 \left(1-e^{-t/RC}\right)$, $(9)$

where $V_0$ is the voltage of the battery. In terms of the voltage across the capacitor,

$V_C(t) = V_0\left(1-e^{-t/RC}\right)$. $(10)$

Eq. 10 also assumes that the capacitor is starting to charge from 0V. We can generalize the charging behavior between an initial voltage $V_{min}$ and final voltage, $V_{max}$, to find the change in the capacitor voltage as

$\Delta V_C(t) = (V_{max} - V_{min}) \left(1-e^{-t/RC}\right)$. $(10b)$

where $\Delta V_C = V_C(t) - V_{min}$.

A graph showing the charging of a capacitor according to the predictions of $(10)$ is shown in Fig. 7. Again, the time constant $\tau= RC$ has meaning. This time, it is the time to reach $1-e^{-1} \approx 63\%$ of its final charge.

Figure 7: Voltage vs. time for a charging capacitor

Observing discharge on the computer

We will use Logger Pro on the computer and the LabQuest Mini (LQM) to measure and plot the voltage across a discharging capacitor.

An RC circuit has been constructed by connecting a 100 k$\Omega$ resistor in parallel with a 10 $\mu$F* capacitor. Clip the red and black ends of the test cable across the RC combination and make sure that the cable is plugged into Channel 1 on the LQM. (See Fig. 8.) Open the file “RC Time Constant” on the computer desktop.

*Capacitor may vary substantially from stated value, $\pm 20\%$ is typical

Figure 8: (Left) The LQM with RC circuit attached (via Channel 1) and the resistor used for charging to +5V (via Channel 2). (Right) The 100 k$\Omega$ + 10 $\mu$F capacitor combination, connected between the clips. Note that the capacitor has a specific orientation and must be connected with the arrows pointing from high voltage (red clip) to low voltage (black clip).

If the file is not there (or is not working because a previous group has edited and saved over it), a fresh copy is available here.

In order to charge the capacitor, we shall make use of the 5 volts provided by one of the unused connectors on the LQM. Insert a resistor into the +5 V output on the LQM as shown in Fig. 9.

Figure 9: +5V out of the LQM connector

Click on the Collect button. Now, momentarily touch the red end of the RC combination to the resistor providing the charging potential (approximately 5 volts). As soon as you remove the RC combination from the charging potential, the computer should begin to display the discharge of the capacitor on the voltage vs. time plot.

Equation (7) predicts an exponential decay of the voltage. We can test this prediction by fitting an exponential to the data. To fit the data, follow these steps:

  • Click on the graph window to select it.
  • From the Analyze menu, select Curve Fit.
  • In the Curve Fit popup window, choose Natural Exponent [A*exp(-C*x)+B] in the General Equation box, and click Try Fit.
  • The text box immediately below the graph will show the results of the fit with the best fit values for the constants A, B, and C.
Qualitatively, are the data well represented by an exponential decay? What is the value of the time constant you get from the fit?

From the Analyze menu, select Examine. A cursor will appear which you can use to read data points directly from the graph.

Using the cursor, measure the time for the voltage to drop to $1/e \approx 0.368$ of its initial value. Compare this time with the time constant predicted by Eq (7) .

Measuring capacitance on a scope

We will now move on to using the oscilloscope to observe (and make measurements of) capacitor charging and discharging.

How to replicate the above measurement with a scope (optional)

We used the computer to record the discharge of the RC circuit above because it was a long time constant (~seconds). When we want to look at something that is happening faster (~microseconds or milliseconds), we need to use an oscilloscope. That said, the scope will still work with long signals.

If you want to get the scope to gather the exact same data as you did with Logger Pro, change the parameters on the scope as follows:

  • Horizontal should be on the order of 1 or 2 seconds
  • Vertical should be on a 2V scale
  • Triggering should be set to channel 1
  • Triggering:Threshold the trigger threshold should be around 4.5 V
  • Triggering:Slope should be set to falling
  • Triggering:Mode should be set to normal

Next, attach the scope leads across the RC components (you may have to add alligator clips to your red and black cables) and charge the capacitor across the +5 V from the LQM just as you did before. It will take several seconds before the scope updates and shows you the full decay.

Wiring the circuit

Pick one capacitor from those provided in the lab and construct the circuit shown in Fig. 10. Wiring in this manner keeps the circuit grounds connected together as shown.

For the rest of the experiment, do not use the joined “resistor + capacitor combination” used above for the Logger Pro demo. We will instead use the resistance box (which can be used to vary $R$) and the small capacitor mounted in the double banana jack (which will have fixed $C$).

A standard capacitor mounted on a double banana jack.

A variable resistance box

Combination resistor and capacitor. DO NOT USE FOR THIS PART.
What's this source resistance thing about?

When working with electronics, test devices like function generators are often modeled as a combination of an ideal voltage/current source in series with some resistance. $50\text{ \(\Omega\)}$ has become a common standard designed into devices because it is near optimal for transferring signal/power over common coaxial cables. If we didn't use output resistance when considering devices then we could end up making predictions like having infinite current in a circuit, which we (thankfully) don't observe.

What's this meter resistance thing about?

When you attach a meter or a scope to a circuit to act like a voltmeter, you don't want the meter to affect the rest of the circuit. Therefore, in order to limit how much current it can draw, a voltmeter often has a larger resistor (on the order of 1 or 10 M$\Omega$ in parallel with the leads.

Figure 10: Capacitor charging circuit. You will need to use wires to make the connections indicated with filled solid circles. Note that the source and meter resistances are an intrinsic part of our devices; you don't have to add them to the circuit yourself.

Do not ingest.

Why not?

These used to be referred to as as “Chiclet” capacitors due to resembling a brand of gum:

A note on electrical grounds

The function generator and scope you will use each have a power cord with a third round prong. Plugging the power cord into a grounded power outlet connects the chassis through the third prong to wires which eventually connect to a cold water pipe or to a copper rod which is buried in the earth. Practically, the earth may be considered an infinite source or sink of electrons. When such a connection is made, the device is said to be grounded. For our purposes, all grounds may be considered to be at the same potential which is defined to be $V = 0$.

A physical analogy for ground

When we talk about grounding a circuit or device, it is usually a shorthand way of saying that there's a voltage somewhere that we're going to arbitrarily label as our starting place, 0V. If our measurement devices aren't set to reference the same point, it'd cause problems.

A similar situation is like how the ground floor of a building might be arbitrary if it is built into an incline, hill, or has strange stairs. If I tell you to come into Kersten and walk up the first set of stairs past the doors to get to your lab room, you'd be fine coming from the North entrance or the lecture hall entrance, but if you came in the South doorway would end up somewhere completely unwanted.

One difference here is that our function generator and oscilloscope will try and set one connection point to ground. The closes physical analogy would be to somebody trying to warp the building geometry to make all the entryways behave the same, which would be a bad time if you're inside at the time.

The round connectors on the front panels of the generator and scope are called BNC connectors. The central pin of the BNC carries the varying voltage while the BNC case is held at the chassis potential which may be grounded.

Setting up the oscilloscope and function generator

Many phenomena of interest in scientific research have characteristic times orders of magnitude shorter than one second, and to observe them we have to use special electronic equipment. The instrument ideally suited for measuring voltages that change rapidly is the oscilloscope. In its most common usage, the oscilloscope produces a plot of voltage as a function of time.

In order to observe the charge and discharge of a capacitor when the $RC$ time is a fraction of second, we have to use a fast electronic switch. The instrument we will use for that purpose is the square wave generator, whose output is shown in Fig. 11.

Figure 11: Square wave

Select a resistance of 1 k$\Omega$ on the resistor box.

Turn on the function generator set to a square wave and adjust the scope to obtain a stable trigger when measuring the voltage across the capacitor. If you have difficulty finding the signal ask the instructor or another lab group for help.

Each time the voltage from the function generator square wave changes, the capacitor will discharge with its characteristic time constant. Using the scope, zoom in on one of the edges of the square wave and adjust the function generator frequency until you can observe a capacitive discharge across the capacitor when the voltage flips. If you have difficulty finding the signal ask the instructor or another lab group for help.

Measure the time constant (and capacitance) of the capacitor

Use the digital scope time cursors to measure the time constant of the capacitor from the voltage decay displayed on the scope. Since you know the resistance value you selected on the resistor box, use the time constant to calculate the capacitance of your unknown capacitor.

Estimate uncertainties in the measured quantities.

Extending your measurement

Next, you will extend your measurement to the extremes.

You will eventually make measurements of the time constant as you change the resistance over the full range (from a few ohms to a few megaohms), but we will split the data taking up into three distinct parts. Think carefully about what values to measure at in each case; you don't have time to do the whole range in 1-ohm increments! Plot your values as you go in the Google Colab notebook below.

Range 1: Middle values

For a range of resistances from 500 $\Omega$ to 50 k$\Omega$, plot your measured time constant (with uncertainty) versus resistance and test the data against the model predicted above: $\tau = RC$.

Do your data fit the model well? Fitting against this model, what do your data suggest is the capacitance of the capacitor, $C$ (with uncertainties)?

Range 2: Low values

As we adjust our resistor box to lower and lower resistances, we should begin to see that the output resistance of the function generator matters. We will call this resistance the source resistance, $R_s$.

If the source resistance is in series with the output of the function generator, then that means it is also in series with the resistance box. Since resistors in series simply add together, we can modify our model to take this into account as $\tau = (R + R_s)C$.

Continue collecting data as you take the resistance to low values, $R < 500~\Omega$. (You can even take it to $R = 0$.) Now try fitting your data to the new model, this time using the value of $C$ you found from your fit above and letting the fit determine the best value for the function generator source resistance, $R_s$.

Range 3: High values

Finally, we can take our resistance up to the highest possible values. As our resistance increases, the internal resistance of the oscilloscope will become important. We will call this resistance the meter resistance, $R_m$.

If the meter resistance is in parallel with the input of the oscilloscope, then that means it is also in parallel with the resistance box (and with the source resistance). Since resistors in parallel add as inverses, we can modify our model to take this into account as $\tau = \frac{(R + R_s)R_m C}{R + R_s + R_m}$.

Continue collecting data as you take the resistance to higher values, $R > 50$ k$\Omega$. Try fitting your data to the new model, using the values of $C$ and $R_s$ which you found from your fits above. This time, we will let the fit determine the best value for the oscilloscope's meter resistance, $R_m$.

"Calculable" capacitor

There are only a few set-ups of this apparatus in the lab.

Calculate $C$ from geometry

Measure the diameter of the capacitor plates and use a micrometer to measure the thickness of the paper spacers shown in Fig. 12, estimating the uncertainties in these quantities.

Try to use only a few small slips of paper… maybe 3 squares, 1 cm x 1 cm. Larger slips will create an uneven gap between plates and too much paper will affect the dielectric constant (to be more that of paper than air).

Figure 12: Calculable capacitor

A calculable capacitor. Also do not ingest.

A micrometer. Each tic mark on the dial is $0.01\text{ mm} = 10^{-5}\text{ m}$, each full revolution is $0.5\text{ mm}$. This unit currently reads $0.75\text{ mm}$. Ingest at your own risk.

Measure the capacitance

Use the same procedure as above to measure the capacitance using the digital scope. Compare the capacitance you expect from calculation to the measurement you made in the lab.

Submit your lab notebook

Make sure to submit your lab notebook by the end of the period. Download a copy of your notebook in PDF format and upload it to the appropriate spot on Canvas. Only one member of the group needs to submit to Canvas, but make sure everyone's name is on the document!

When you're finished, don't forget to log out of both Google and Canvas, and to close all browser windows before leaving!

Post-lab assignment

Answer the questions/prompts below in a new document (not your lab notebook) and submit that as a PDF to the appropriate assignment on Canvas when you are done. You should write the answers to these questions by yourself (not in a group), though you are welcome to talk to your group mates to ask questions or to discuss.


The conclusion is your interpretation and discussion of your data.

  • What do your data tell you?
  • How do your data match the model (or models) you were comparing against, or to your expectations in general? (Sometimes this means using the $t^{\prime}$ test, but other times it means making qualitative comparisons.)
  • Were you able to estimate uncertainties well, or do you see room to make changes or improvements in the technique?
  • Do your results lead to new questions?
  • Can you think of other ways to extend or improve the experiment?

In about one or two paragraphs, draw conclusions from the data you collected today. Address both the qualitative and quantitative aspects of the experiment and feel free to use plots, tables or anything else from your notebook to support your words. Don't include throw-away statements like “Looks good” or “Agrees pretty well”; instead, try to be precise.

REMINDER: Your post-lab assignment is due 24 hours after your lab concludes. Submit a single PDF on Canvas.